We present the results of a lattice study of the normalization constants and second moments of the light -cone distribution amplitudes of longitudinally and transversely polarized rho mesons. The calculation is performed using two flavors of dynamical clover fermions at lattice spacings between 0.060 fm and 0.081 fm, different lattice volumes up to m(pi)L = 6.7 and pion masses down to in, m(pi) = 150 MeV. Bare lattice results are renormalized non-perturbatively using a variant of the RI' -MOM scheme and converted to the (MS) over bar scheme. The necessary conversion coefficients, which are not available in the literature, are calculated. The chiral extrapolation for the relevant decay constants is worked out in detail. We obtain for the ratio of the tensor and vector coupling constants f(rho)(T)/f(p)= 0.629(8) and the values of the second Gegenbauer moments alpha(parallel to)(2) = 0.132(27) and alpha(parallel to)(2) = 0.101(22) at the scale mu = 2 GeV for the longitudinally and transversely polarized rho mesons, respectively. The errors include the statistical uncertainty and estimates of the systematics arising from renormalization. Discretization errors cannot be estimated reliably and are not included. In this calculation the possibility of rho -> pi pi decay at the smaller pion masses is not taken into account.